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David Terr
Ph.D. Math, UC Berkeley

 

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4.5. Graphs of Sine and Cosine Functions

Lecture on How to Graph sin x

 

In the last three sections, we have investigated some of the basic properties of the six trigonometric functions at length. But we still have not graphed these functions. What do their graphs look like? In this section we investigate the graphs of sine and cosine, the two most basic trigonometric functions.

As mentioned at length in Chapter 1, when graphing functions, it is first a good idea to have a table of lots of values of these functions. Thus, we will start by constructing a table of values of sin x and cos x for as many values of x from 0 to 2π as we can calculate. (Since sine and cosine are both periodic with period 2π, we know that their graphs will repeat whenever x varies by a multiple of 2π. For this reason, it is only necessary to tabulate values in one period of these functions.)

We may use Equations (4.4.3) - (4.4.8) to extend Table 4.3.1 to include angles up to 2π. Sparing the details of these calculation, we obtain the following table of values of sin x and cos x. Approximate as well as exact values of all quantities are listed.

x sin x cos x
0.000
0.000
1.000
π/6 ≈ 0.524
1/2 = 0.500
3/2 ≈ 0.866
π/4 ≈ 0.785
2/2 ≈ 0.707
2/2 ≈ 0.707
π/3 ≈ 1.047
3/2 ≈ 0.866
1/2 = 0.500
π/2 ≈ 1.571
1.000
0.000
2π/3 ≈ 2.094
3/2 ≈ 0.866
-1/2 = -0.500
3π/4 ≈ 2.356
2/2 ≈ 0.707
-√2/2 ≈ -0.707
5π/6 ≈ 2.618
1/2 = 0.500
-√3/2 ≈ -0.866
π ≈ 3.142
0.000
-1.000
7π/6 ≈3.665
-1/2 = -0.500
-√3/2 ≈ -0.866
5π/4 ≈ 3.927
-√2/2 ≈ -0.707
-√2/2 ≈ -0.707
4π/3 ≈ 4.189
-√3/2 ≈ -0.866
-1/2 = -0.500
3π/2 ≈ 4.712
-1.000
0.000
5π/3 ≈ 5.236
-√3/2 ≈ -0.866
1/2 = 0.500
7π/4 ≈5.498
-√2/2 ≈ -0.707
2/2 ≈ 0.707
11π/6 ≈5.760
-1/2 = -0.500
3/2 ≈ 0.866
2π ≈ 6.283
0.000
1.000

Table 4.5.1

Next we graph these points and plot smooth curves through them, obtaining the following graphs. Note that the y-scale of these graphs has been exaggerated for clarity. We also continue the graphs for one period to the left of the points listed.

graph of sin x

Figure 4.5.1: Graph of sin x

 

graph of cos x

Figure 4.5.2: Graph of cos x

 

Upon examining these graphs, one notices a few interesting things. For one thing, it is clear that the graph of sin x is symmetric about the origin, implying that sin x is an odd function, while the graph of cos x is symmetric about the y-axis, implying that cos x is an even function. It is in fact easy to prove these assertions, namely

  • (4.5.3) sin(-x) = -sin x
  • (4.5.4) cos(-x) = cos x

We leave the proof of these equations as an exercise.

Another thing to notice is that the graphs of cos x and sin x have the same shape, the graph of cos x merely being shifted along the x-axis to the left by π/2. This shape is known as a sine wave or sinusoid. Mathematically, this fact amounts to the following formula:

  • (4.5.5) sin(x + π/2) = cos x.

This is also easy to prove, using what we already know about sin x and cos x. Once again, we leave the proof as an exercise.

 

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